LAUSR

The triple correlation [Formula: see text] for arbitrary coefficients [Formula: see text] is estimated on average over [Formula: see text] in some short intervals. By introducing a device of short intervals, we are able to reduce this problem to uniform oscillations of one of the three coefficients, say [Formula: see text], against additive characters of [Formula: see text] over short intervals. The argument is simple, but refines previous arguments in certain cases. More precise estimates are also obtained by taking [Formula: see text] to be Fourier coefficients of cusp forms and Möbius functions, which substantially improve previous results.